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In recent years, under the impact of various social, scientific, and technological forces, there has been a widening and intensified awareness of the important role played by mathematics in contemporary culture. Mathematics functions not only as a computational tool, where its power has been increased many fold by utilization of electronic devices, but even more importantly as a means of furnishing abstract conceptual and deductive systems which serve to coalesce and resolve a host of diverse problems, both practical and theoretical.

Although an appreciation of the broadened role of mathematics is widespread, a first-hand acquaintance with the revolutionary changes in mathematical content and approach, which have been fashioned by modern research, have been hitherto reserved for the trained mathematician who devoted many years to technical study; to most laymen these details have remained a complete mystery.

The realization of the immensity of the gap between traditional and contemporary mathematical outlook has naturally led to an increased concern with the kind of mathematics which is, or should be, taught in the schools--from the elementary grades through the colleges--and with the nature of the preparation of those teachers who are entrusted with this vital educational task. In the United States this concern has manifested itself in several ways, among which has been the organization by the National Science Foundation of a series of Summer Institutes for Teachers of Mathematics, conducted at universities throughout the country. Institutes are designed to expose teachers of mathematics from the high schools and colleges to some of the new ideas that have been developed by creative mathematicians.

This book is the outcome of our experiences in teaching and directing three such institutes held at the University of Wyoming. The material of the book is an elaboration of the content of a course of lectures, entitled "Foundations of Mathematics," which was developed by Professor Henkin especially for these institutes.

The warm reception accorded this course by the participants in these . . .